Add $ \dfrac{2}{3-2i} $ and $ 2 $ to get $ \dfrac{ \color{purple}{ -4i+8 } }{ -2i+3 }$.

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{-2i+3}$.

$$ \begin{aligned} \frac{2}{3-2i} +2 & \xlongequal{\text{Step 1}} \frac{2}{3-2i} + \frac{2}{\color{red}{1}} = \frac{ 2 }{ 3-2i } + \frac{ 2 \cdot \color{blue}{ \left( -2i+3 \right) }}{ 1 \cdot \color{blue}{ \left( -2i+3 \right) }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2 } }{ 3-2i } + \frac{ \color{purple}{ -4i+6 } }{ 3-2i }=\frac{ \color{purple}{ -4i+8 } }{ -2i+3 } \end{aligned} $$

Divide $ \, 8-4i \, $ by $ \, 3-2i \, $ to get $\,\, \dfrac{32+4i}{13} $.
( view steps )